Parametrizing recollement data

0.44 MB·

by  Pedro Nicolas

PARAMETRIZING RECOLLEMENT DATA
PEDRONICOLA´SANDMANUELSAOR´IN
8
0
Abstract. Wegiveageneralparametrization ofallthe recollementdatafor
0
atriangulatedcategorywithasetofgenerators. Fromthiswededuceachar-
2
acterization of when a perfectly generated (or aisled) triangulated category

n is a recollement of triangulated categories generated by a single compact ob-
a ject. Also, we use homological epimorphisms to give a complete and explicit
J description of all the recollement data for (or smashingsubcategories of) the
3 derived category of a k-flat dg category. In the final part we give a bijec-
tionbetweensmashingsubcategoriesofcompactlygeneratedtriangulatedcat-

] egories andcertainidealsofthe subcategory ofcompactobjects, inthespirit
T of H.Krause’swork[33]. Thisbijectionimpliesthe followingweakversionof
R the generalized smashing conjecture: in a compactly generated triangulated
category everysmashingsubcategory isgenerated byasetofMilnorcolimits
.
h ofcompactobjects.
t
a
m
[

1 1. Introduction
v
1.1. Motivations. The definition of t-structure and recollement for triangulated
0
0 categories was given by A. A. Beilinson, J. Bernstein and P. Deligne in their work
5 [2] on perverse sheaves. The notion of t-structure is the analogue of the notion
0 of torsion pair [10, 47, 4] for abelian categories. Accordingly, the ‘triangulated’
.
1 analogue of a torsion torsionfree(=TTF) triple [20, 47, 4], still called TTF triple,
0 consists of a triple ( , , ) such that both ( , ) and ( , ) are t-structures. In
8 general,TTFtriplesXonYatZriangulatedcategorXy YareinbYijecZtionwith(equivalence
0 D
classes of) ways of expressing as a recollement of triangulated categories, and
: D
v with smashing subcategories of for instance when it is perfectly generated.
Xi One of the aims of this paperDis to give a parametrizationof all the TTF triples
on triangulated categories of a certain type including those which are well (or
r
a even perfectly) generated. This parametrization might result naive but neverthe-
less, together with B. Keller’s Morita theory for derived categories [23], it yields
a generalization of some results of [11, 21] and offers an unbounded and abstract
version of S. K¨onig’s theorem [30] on recollements of right bounded derived cate-
goriesofalgebras. Ina forthcomingpaper we will study the problemofdescending
the parametrizationfrom unbounded to right bounded derived categories, and the
corresponding lifting.
Date:December18,2007.
1991 Mathematics Subject Classification. 18E30,18E40.
Key words and phrases. Compact object, derived category, dg category, homological epimor-
phism,perfectobject,recollement,smashing,t-structure,torsionpair,triangulatedcategory.
TheauthorshavebeenpartiallysupportedbyresearchprojectsfromtheD.G.I.oftheSpanish
Ministryof Education and the Fundacio´n S´eneca of Murcia, with a part of FEDER funds. The
firstauthor hasbeenalsosupportedbytheMECDgrantAP2003-2896.
1
2 PEDRONICOLA´SANDMANUELSAOR´IN
The following facts suggest that, in the case of derived categories, a more so-
phisticated parametrization is possible:
1) TTFtriplesoncategoriesofmodulesarewellunderstoodandatangibleparam-
etrization of them was given by J. P. Jans [20].
2) A natural proof of J. P. Jans’ theorem uses P. Gabriel’s characterization of
categories of modules among abelian categories [12], which is at the basis of
Morita theory.
3) P.Gabriel’scharacterizationadmitsa‘triangulated’analoguewhichwasproved
by B. Keller, who developed a Morita theory for derived categories of dg cate-
gories in [23] (and later in [24, 25],...).
4) It seems that derived categories of dg categories play the rˆole, in the theory
of triangulated categories, that module categories play in the theory of abelian
categories.
Then, another aim of this paper is to give a touchable parametrization of TTF
triples on derived categories of dg categories by using B. Keller’s theory, and to
elucidate their links with H. Krause’s parametrization[31, 33] of smashing subcat-
egories of compactly generated triangulated categories. For this, we use a general-
izationof the notion ofhomological epimorphism due to W. Geigle and H. Lenzing
[13]. Homological epimorphisms appear as (stably flat) universal localizations in
the work of P. M. Cohn [9], A. H. Schofield [45], A. Neeman and A. Ranicki [39],
...Recently,H.Krausehasstudied[33]thelinkbetweenhomologicalepimorphisms
ofalgebrasand: thechain map lifting problem, the generalized smashing conjecture
and the existence of long exact sequences in algebraic K-theory. Homological epi-
morphisms also appear in the work of L. Angeleri Hu¨gel and J. Sa´nchez [1] on the
construction of tilting modules induced by ring epimorphisms.
1.2. Contents. In section 2, we fix some terminology and recall some results on
triangulatedcategories,emphasizing B.Keller’swork[23,25] onderivedcategories
of dg categories. We prove, however, some apparently new results which measure
the distance between “compact” and: ‘self-compact’, “perfect”, “superperfect”. In
section3,weintroducethenotionofrecollement-definingclass (subsection3.1)and
prove how to find recollement-defining sets in aisled triangulated categories (sub-
section 3.2) and in perfectly generated triangulated categories (subsection 3.3).
In subsection 3.4, recollement-defining sets enable us to parametrize all the TTF
triples on a triangulated category with a set of generators, and all the ways of
expressing a ‘good’ triangulated category as a recollement of compactly generated
triangulated categories. In section 4, we introduce the notion of homological epi-
morphisms of dg categories, generalizingthe homologicalepimorphisms of algebras
of W. Geigle and H. Lenzing [13]. We easily prove that this kind of morphisms al-
waysinduce a TTF triple, whichallows us to giveseveralexamples of recollements
for unbounded derived categories of algebras which were already known for right
bounded derived categories (cf. S. K¨onig’s paper [30]). Conversely, we prove that
every TTF triple on the derived categoryof a k-flat dg category is induced by a
A
homologicalepimorphismstarting in . This correspondence between TTF triples
A
andhomologicalepimorphismskeepsalotofsimilitudeswiththeoneaccomplished
byJ.P.Jans[20]formodule categories. Insection5,westate aparametrizationof
smashing subcategories of a compactly generated algebraic triangulated category
whichusesthemainresultsofsubsection3.4andsection4. Finally,insection6,we
PARAMETRIZING RECOLLEMENT DATA 3
analysehowidempotent two-sidedideals ofthe subcategory c ofcompactobjects
D
appear in the description of TTF triples on a compactly generated triangulated
category . More concretely, in Theorem 6.2 we prove that an idempotent two-
D
sided ideal of c, which is moreover stable under shifts in both directions, always
D
induces a nicely described TTF triple on . This, together with assertion 2’) of
D
H. Krause’s [31, Theorem 4.2], gives a short proof of a result (cf. Theorem 6.4.2)
in the spirit of H. Krause’s bijection [33, Theorem 11.1, Theorem 12.1, Corollary
12.5 and Corollary 12.6] between smashing subcategories and special idempotent
two-sidedideals. As a consequence (cf. Corollary6.4.2), we get the following weak
version of the generalized smashing conjecture: every smashing subcategory of a
compactly generatedtriangulatedcategoryis generatedby a set of Milnor colimits
of compact objects. Another consequence (cf. Corollary 6.4.3) is that, when is
D
algebraic,we recover precisely H. Krause’s bijection.
1.3. Acknowledgements. We are grateful to Bernhard Keller for several discus-
sions concerning dg categories.
2. Notation and preliminary results
Unless otherwise stated, k will be a commutative (associative, unital) ring and
every additive category will be assumed to be k-linear. We denote by Modk the
category of k-modules. Given a class of objects of an additive category , we
Q D
denote by ⊥D (or ⊥ if the category is clearly assumed) the full subcategory
Q Q D
of formed by the objects M whichare right orthogonal to every object of , i.e.
D Q
suchthat (Q,M)=0 for allQ in . Dually for ⊥D . When is a triangulated
D Q Q D
category,theshiftfunctor willbedenotedby?[1]. Whenwespeakof“alltheshifts”
or“closedunder shifts” andso on, we willmean“shifts inboth directions”,that is
to say, we will refer to the nth power ?[n] of ?[1] for all the integers n Z. In case
∈
we want to consider another situation (e.g. non-negative shifts ?[n], n 0) this
≥
will be said explicitly. We will use without explicit mention the bijection between
t-structures ona triangulatedcategory andaisles in , provedby B. Keller and
D D
D. Vossieck in [29]. If ( , [1]) is a t-structure on a triangulated category , we
U V D
denote by u : ֒ and v : ֒ the inclusion functors, by τ a right adjoint
U
U → D V → D
to u and by τV a left adjoint to v.
2.1. TTF triples and recollements. A TTF triple on is a triple ( , , ) of
D X Y Z
full subcategories of such that ( , ) and ( , ) are t-structures on . Notice
D X Y Y Z D
that, in particular, , and are full triangulatedsubcategories of . It is well
X Y Z D
known that TTF triples are in bijection with (equivalence classes of) recollements
(cf. [2, 1.4.4], [38, subsection 9.2], [41, subsection 4.2]). For the convenience of the
reader we recall here how this bijection works. If
i∗ j∗
{{ i∗ // {{ j∗ //
F U
D cc Dcc D
i! j!
expresses as a recollement of and , then
F U
D D D
(j( ),i ( ),j ( ))
! U ∗ F ∗ U
D D D
4 PEDRONICOLA´SANDMANUELSAOR´IN
is a TTF triple on , where by j( ) we mean the essential image of j, and
! U !
D D
analogouslywiththe otherfunctors. Conversely,itis straightforwardtocheckthat
if ( , , ) is a TTF triple on , then is a recollement of and as follows:
X Y Z D D Y X
τY zτZx
|| y // || τX // ,
Y bb Dbb X
τY x
x τZ z
Notice that for a TTF triple ( , , ) the compositions and
τX are mutually quasi-inveXrseYtrZiangle equivalences (cXf. [→41,DLe→mmZa 1.6.7Z]).→
D→X
2.2. Generators and infinite d´evissage. Let be a triangulated category. We
D
saythatitisgenerated byaclass ofobjectsifanobjectM of iszerowhenever
Q D
(Q[n],M)=0
D
for every object Q of and every integer n Z. In this case, we say that is a
Q ∈ Q
class of generators of and that generates .
D Q D
If hassmallcoproducts,givenaclass ofobjectsof wedenotebyTria ( )
D
D Q D Q
(or Tria( ) if the category is clear) the smallest full triangulated subcategory
Q D
of containing and closed under small coproducts. We say that satisfies the
D Q D
principle of infinite d´evissage with respect to a class of objects if it has small
Q
coproducts and =Tria( ). In this case, it is clear that is generated by .
D Q D Q
Conversely, the first part of the following lemma states that under certain hy-
pothesis ‘generators’implies ‘d´evissage’.
Lemma. 1) Let be a triangulated category with small coproducts and let ′ be
D D
a full triangulated subcategory generated by a class of objects . If Tria( ) is
Q Q
an aisle in contained in ′, then ′ =Tria( ).
D D D Q
2) Let be a triangulated category and let ( , ) be a t-structure on with tri-
D X Y D
angulated aisle.
2.1) If is a class of generators of , then τY( ) is a class of generators of
Q D Q
.
Y
2.2) A class of objects of generates if and only if the objects of are
Q X X Y
precisely those which are right orthogonal to all the shifts of objects of .
Q
Proof. 1) Given an object M of ′ there exists a triangle
D
M′ M M′′ M′[1]
→ → →
with M′ in Tria( ) and M′′ in Tria( )⊥D. Since M′ and M are in ′, then so is
Q Q D
M′′. But ′ is generated by , which implies that M′′ = 0 and so M belongs to
D Q
Tria( ).
Q
2) Left as an exercise. √
2.3. (Super)perfectnessandcompactness. AnobjectP of isperfect (respec-
D
tively,superperfect)ifforeverycountable(respectively,small)familyofmorphisms
M N , i I, of such that the natural morphism M N exists the
i → i ∈ D I i → I i
induced map ` `
(P, M ) (P, N )
i i
D →D
aI aI
is surjective provided every map
(P,M ) (P,N )
i i
D →D
PARAMETRIZING RECOLLEMENT DATA 5
is surjective. Particular cases of superperfect objects are compact objects, i.e.
objects P such that (P,?) commutes with small coproducts.
D
The following lemma is very useful. It shows some links between t-structures
and(super)perfect and compactobjects. First we remind the followingdefinitions.
Let be a triangulated category. A contravariant functor H : Modk is
D D →
cohomological if for every triangle
f g
L M N L[1]
→ → →
the sequence
H(g) H(f)
H(N) H(M) H(L).
→ →
is exact. We say that satisfies the Brown’s representability theorem for cohomol-
ogy ifeverycohomologDicalfunctorH : Modktakingsmallcoproductstosmall
D →
products is representable.
Lemma. 1) Let be a triangulated category and let ( , ) be a t-structure on
D X Y D
with triangulated and such that the inclusion functor y : ֒ preserves
X Y → D
small coproducts. If M is a perfect (respectively, superperfect, compact) object
of , then τYM is a perfect (respectively, superperfect, compact) object of .
D Y
2) The adjoint functor argument: Let be a triangulated category with small co-
D
products and let ′ be a full triangulated subcategory of closed under small
D D
coproducts and satisfying the Brown’s representability theorem. In this case ′
D
is an aisle in . In particular, if is a set of objects of which are perfect in
D P D
Tria( ), then Tria( ) is an aisle in .
P P D
Proof. 1)iswellknown[37, Lemma 2.4]forthe case ofcompactobjects. Using the
adjunction (y,τY) it also follows easily for the case of (super)perfect objects.
2) If
ι: ′ ֒
D →D
is the inclusion functor, for an object M of we define the functor
D
H(?):= (ι(?),M): ′ Modk,
D D →
which takes triangles to exact sequences and coproduct to products. Then, by
hypothesis this functor is represented by an object, say τ(M) ′. By Yoneda
∈ D
lemma it turns out that the map M τ(M) underlies a functor ′ which
7→ D → D
is right adjoint to ι. Therefore, by [29, subsection 1.1] we have that ′ is an
D
aisle. Finally, if is a set of objects of which are perfect in Tria( ), then by
P D P
H. Krause’s theorem [32, Theorem A] we know that Tria( ) satisfies the Brown’s
P
representability theorem. √
Remark. By using Lemma 2.2 and the lemma above we have the following: if a
triangulated category with small coproducts is generated by a set of objects
D P
such that Tria( ) is an aisle in (e.g. if the objects of are perfect in Tria( )),
P D P P
then it satisfies the principle of infinite d´evissage with respect to that set, i.e.
=Tria( ).
D P
A triangulated category with small coproducts is perfectly (respectively, super-
perfectly, compactly) generated if it is generated by a set of perfect (respectively,
superperfect,compact)objects. ATTFtriple( , , )onatriangulatedcategory
X Y Z
withsmallcoproductsisperfectly (respectively,superperfectly,compactly)generated
if so is as a triangulated category.
X
6 PEDRONICOLA´SANDMANUELSAOR´IN
2.4. Smashing subcategories. Let be a triangulated category with small co-
D
products. Asubcategory of is smashing ifitis afulltriangulatedsubcategory
X D
of which, moreover,is an aisle in whose associated coaisle ⊥ is closed under
D D X
smallcoproducts. Itisprovedin[41]thatthis agreeswithprobablymorestandard
definitions of “smashing subcategory”.
Smashingsubcategoriesallowtotransferlocalphenomenatoglobalphenomena:
Lemma. IfM isaperfect(respectively, superperfect,compact)objectofasmashing
subcategory of a triangulated category with small coproducts, then M is perfect
X D
(respectively, superperfect, compact) in .
D
Proof. Usethatasmallcoproductoftrianglesassociatedtothet-structure( , ⊥)
X X
is again a triangle associated to this t-structure. √
Nowwefixthenotationforaparticularkindofconstructionwhichwillbecrucial
at certain steps. Let be a triangulated category and let
D
M f0 M f1 M ...
0 1 2
→ → →
beasequenceofmorphismsof suchthatthecoproduct M existsin . The
Milnor colimit of this sequenceD, denoted by McolimMn,`isng≥iv0en,nup to non-Dunique
isomorphism, by the triangle
M 1−σ M π McolimM M [1],
n n n n
→ → →
na≥0 na≥0 na≥0
where the morphism σ has components
M fn M can M .
n n+1 m
→ →
ma≥0
The above triangle is said to be the Milnor triangle (cf. [25, 36]) associated to the
sequence f , n 0. The notion of Milnor colimit has appeared in the literature
n
≥
under the name of homotopy colimit (cf. [5, Definition 2.1], [38, Definition 1.6.4])
and homotopy limit (cf. [23, subsection 5.1]). However, we think it is better to
keep this terminology for the notions appearing in the theory of derivators (cf.
[34, 35, 8]).
Proposition. Let be a triangulated category with small coproducts and let P be
D
an object of . The following conditions are equivalent:
D
1) P is compact in .
D
2) P satisfies:
2.1) P is perfect in .
D
2.2) P is compact in the full subcategory Sum( P[n] ) of formed by small
n∈Z
{ } D
coproducts of shifts of P.
2.3) Tria(P)⊥ is closed under small coproducts.
3) P satisfies:
3.1) P is compact in Tria(P).
3.2) Tria(P)⊥ is closed under small coproducts.
4) P satisfies:
4.1) P is superperfect in .
D
4.2) P is compact in Sum( P[n] ).
n∈Z
{ }
PARAMETRIZING RECOLLEMENT DATA 7
Proof. 2) 1) If P is perfect in , by Theorem A of [32] and the adjoint functor
⇒ D
argument(cf. Lemma2.3)weknowthat =Tria(P)isanaislein . Assumption
X D
2.3) says that is a smashing subcategory. Thanks to Lemma 2.4, it suffices to
X
prove that P is compact in . For this, we will use the following facts:
X
a) Every object of is the Milnor colimit McolimX of a sequence
n
X
X f0 X f1 X f2 ...
0 1 2
→ → →
where X as well as each mapping cone Cone(f ) is in Sum( P[n] ).
0 m n∈Z
{ }
b) ThankstotheproofofTheoremAof[32](cf.alsotheproofof[46,Theorem2.2]),
if X ,f is a direct system as in a) we know that the natural morphism
n n n≥0
{ }
lim (P,X ) (P,McolimX )
n n
D →D
−→
is an isomorphism.
c) Hypothesis 2.2) implies that, for any fixed natural number m 0, the functor
≥
(P,?) preserves small coproducts of m-fold extensions of objects of the class
D
Sum( P[n] ).
n∈Z
{ }
Let McolimXi , i I, be an arbitrary family of objects of . Then the natural
n ∈ X
morphism
(P,McolimXi) (P, McolimXi)
X n →X n
ai∈I ai∈I
is the composition of the following natural isomorphisms
(P,McolimXi)= lim (P,Xi)= lim (P,Xi)=
X n ∼ X n ∼ X n ∼
ai∈I ai∈In−→≥0 n−→≥0ai∈I
= lim (P, Xi)= (P,Mcolim Xi)= (P, McolimXi)
∼ X n ∼X n ∼X n
n−→≥0 ai∈I ai∈I ai∈I
Hence, P is compact in the smashing subcategory .
X
3) 1) By the adjoint functor argumentwe knowthat Tria(P) is an aisle in ,
⇒ D
andcondition3.2)ensuresthat, moreover,itis a smashingsubcategory. Hence the
lemma above finishes the proof.
4) 1) Of course, 4) implies 2), and 2) implies 1). However, there exists a
⇒
shorter proof pointed out by B. Keller. Let M , i I, be a family of objects of
i
∈
and take, for each index i I, an object Q Sum( P[n] ) together with a
i n∈Z
D ∈ ∈ { }
morphism Q M such that the induced morphism
i i
→
(P,Q ) (P,M )
i i
D →D
is surjective. Since P is superperfect, this implies that the morphism
(P, Q ) (P, M ).
i i
D →D
aI aI
is surjective. Now consider the commutative square
(P, Q ) // (P, M )
D OO I i D OO I i
` `
≀ can can
(P,Q ) // (P,M )
ID i ID i
` `
The first vertical arrow is an isomorphism by assumption 4.2, and the horizontal
arrowsaresurjections. Then,thesecondverticalarrowissurjective. But,ofcourse,
it is also injective, and so bijective. √
8 PEDRONICOLA´SANDMANUELSAOR´IN
The following result is a consequence of Lemma 2.2 and Lemma 2.3.
Corollary. If is a perfectly generated triangulated category, then smashing sub-
D
categories of are in bijection with TTF triples on via the map
D D
( , ⊥,( ⊥)⊥).
X 7→ X X X
Proof. Indeed,if( , , )isaTTFtriple,then isasmashingsubcategorysince
X Y Z X
being an aisle is always closed under coproducts. Conversely,if is a smashing
Y X
subcategory,then ( , ) is a t-structure on . But now, by using Lemma 2.2 and
X Y D
Lemma2.3wehavethatτY takesthesetofperfectgeneratorsof toasetofperfect
D
generators . Therefore, is a perfectly generated triangulated category closed
Y Y
under small coproducts in , and by the adjoint functor argument we conclude
D
that is an aisle. √
Y
2.5. B. Keller’s Morita theory for derived categories. Let be a small dg
A
category (cf. [23, 27]). It was proved by B. Keller [23] that its derived category
is a triangulated category compactly generated by the modules A∧ := (?,A)
DA A
represented by the objects A of . Conversely, he also proved [23, Theorem 4.3]
A
that every algebraic triangulated category (namely, a triangulated category which
is triangle equivalent to the stable categoryof a Frobeniuscategory[17, 15, 28, 14,
24]) with small coproducts and with a set of compact generators is the derived
P
category of a certain dg category whose set of objects is equipotent to .
P
The proof of Theorem 4.3 of [23] has two parts.
First part: it is proved that every algebraic triangulated category admits an
enhancement, i.e. comes from an exact dg category. We say that a dg category
′ is exact or pretriangulated [26, 27] if the image of the (fully faithful) Yoneda
A
functor
Z0 ′ ′, M M∧ := ′(?,M)
A →CA 7→ A
is stable under shifts and extensions (in the sense of the exact structure on in
CA
which the conflations are the degreewise split short exact sequences). If ′ is an
A
exact dg category, then Z0 ′ becomes a Frobenius category and Z0 ′ = H0 ′ is
A A A
a full triangulated subcategory of ′. B. Keller has shown [26, Example 2.2.c)]
HA
thatif is a Frobenius categorywith class of conflations , then =H0 ′ for the
C E C A
exact dg category ′ formed by the acyclic complexes with -projective-injective
A E
components over .
C
Second part: it proves the following.
Proposition. Let ′ be an exact dg category such that the associated triangulated
A
category H0 ′ is compactly generated by a set of objects. Consider as a dg
A B B
category, regarded as a full subcategory of ′. Then, the map
A
M M∧ := ′(?,M)
7→ |B A |B
induces a triangle equivalence
H0 ′ ∼ .
A →DB
The dg category associated to the Frobenius category in the first part of the
proofof[23,Theorem4.3]is notveryexplicit. However,manytimes inpracticewe
are already like in the second step of the proof, which allows us a better choice of
thedgcategory. Inwhatfollows,wewillrecallhowthisbetter choicecanbe made.
Let be a set ofobjects of and define as the dg subcategoryof the exact
P DA B
dg category (cf. [27] for the notation) formed by the -injective or fibrant
dg
C A H
PARAMETRIZING RECOLLEMENT DATA 9
resolutions iP [23, 27] of the modules P of . Then we have a dg - -bimodule
P B A
X defined by
X(A,B):=B(A)
for A in and for B in ,and we have a pair (? X, om (X,?)) of adjoint dg
B A
A B ⊗ H
functors
dg
COO A
?⊗BX HomA(X,?)
(cid:15)(cid:15)
dg
C B
For instance, om (X,?) is defined by om (X,M) := ( )(?,M) for M
H A H A CdgA |B
in . These functors induce a pair of adjoint triangle functors between the
dg
C A
corresponding categories up to homotopy [23, 27]
HOO A
?⊗BX HomA(X,?)
(cid:15)(cid:15)
.
HB
The total right derived functor R om (X,?) is the composition
A
H
i ֒ HomA(X,?) ,
i
DA→HA →HA → HB →DB
where i is the -injective resolution or fibrant resolution functor [23, 27], and the
H
total left derived functor ? LX is the composition
⊗B
p ֒ ?⊗BX ,
p
DB →H A →HB → HA→DA
wherepisthe -projective resolution orcofibrant resolution functor[23,27]. They
H
form a pair of adjoint triangle functors at the level of derived categories
DOO A
?⊗LBX RHomA(X,?)
(cid:15)(cid:15)
DB
The following is an easy consequence of Proposition 2.5.
Corollary. Assume that the objects of are compact in the full triangulated sub-
P
category Tria( ) of . Then:
P DA
1) the functors (? L X,R om (X,?)) induce mutually quasi-inverse triangle
⊗B H A
equivalences
RHomA(X,?)
//
Tria( )
P oo ?⊗LX DB
B
which gives a bijection between the objects of and the -modules B∧ repre-
P B
sented by the objects B of ,
B
2) Tria( ) is an aisle in with truncation functor given by the map
P DA
M R om (X,M) LX.
7→ H A ⊗B
10 PEDRONICOLA´SANDMANUELSAOR´IN
∼
Proof. 1) Since the -injective resolution functor i : is a triangle
i
H DA → HA
equivalence, it induces a triangle equivalence between Tria( ) and a certain full
P
triangulated subcategory of . This subcategory is algebraic, i.e. the stable
i
H A
category of a certain Frobenius category , because it is a subcategory of .
C C HA
Since is the H0-category of the exact dg category , then is the H0-
dg
HA C A C
category of an exact dg subcategory of . Moreover, is compactly generated
dg
C A C
by the objects of . Then, by Proposition 2.5 the restriction of om (X,?) to
A
B H C
induces a triangle equivalence
HomA(X,?) .
C → HB →DB
The picture is:
H0( )=
dg
C AOO HA
(cid:31)?
i
//
DAOO ∼ HiOOA
(cid:31)? (cid:31)?
Tria( ) // //
P ∼ C ∼ DB
ThecompositionofthebottomarrowsistherestrictionofR om (X,?)toTria( ).
A
H P
Notice that by the adjoint functor argument (cf. Lemma 2.3) Tria( ) is an aisle,
P
and so Tria( ) = ⊥(Tria( )⊥). Finally, by adjunction the image of ? LX is in
P P ⊗B
⊥(Tria( )⊥) = Tria( ). Alternatively, we can use that satisfies the principle
P P DB
of infinite d´evissage with respect to the modules B∧ , B , to deduce that the
∈ B
image of ? LX is contained in Tria( ). Then, we can prove that
⊗B P
? LX : Tria( )
⊗B DB → P
is a triangle equivalence by using [23, Lemma 4.2].
2) Since ? L X : is fully faithful, the unit η of the adjunction
⊗B DB → DA
(? L X,R om (X,?)) is an isomorphism. Then, when we apply the functor
⊗B H A
? LX R om (X,?) to the counit δ we get an isomorphism. This shows that
⊗B ◦ H A
for each module M in , the triangle of
DA DA
R om (X,M) LX δM M M′ +
H A ⊗B → → →
satisfies that R om (X,M′) L X = 0. Since ? L X is fully faithful, then
H A ⊗B ⊗B
R om (X,M′)=0. That is to say,
A
H
R om (X,M′)(P)=( )(P,iM′)
A dg
H C A
is acyclic for each object P in . But then, we have
P
Hn(CdgA)(P,iM′)=(HA)(P,iM′[n])∼=(DA)(P,M′[n])=0
for each object P of and each integer n Z. This implies, by infinite d´evissage,
P ∈
that M′ belongs to the coaisle Tria( )⊥ of Tria( ). √
P P
Remark. This result generalizes [21, Theorem 1.6] and [11, Theorem 2.1]. In-
deed, if is the dg category associated to the dg k-algebra A and the set
A P
has only one element P, then is the dg category associated to the dg algebra
B